
Given a tensor \({\cal Z}\in \mathbb {R}^{n_1\times \cdots \times n_d}\) , the orthogonal projection of \({\cal Z}\)  onto \(T_{{\cal X}} {\cal M}_{\mathbf {r}} \)  is [1]}
\({\cal P}_{T_{{\cal X}}{\cal M}_{\mathbf {r}}}\left({\cal Z}\right)= \sum _{k = 1}^d \phi \left(\left\lbrace {\cal X}^1,\cdots ,\delta {\cal Z}^k,\cdots ,{\cal X}^d\right\rbrace \right),\) 
